Synthesis and Characterization of Core–Double-Shell-Structured PVDF-grafted-BaTiO3/P(VDF-co-HFP) Nanocomposite Films

Core–double-shell-structured nanocomposite films consisting of polyvinylidene fluoride-grafted-barium titanate (PVDF-g-BT) incorporated into a P(VDF-co-hexafluoropropylene (HFP)) copolymer matrix were produced via a solution mixing method for energy storage applications. The resulting films were thoroughly investigated via spectroscopic, thermal, and morphological analyses. Thermogravimetric data provided an enhancement of the thermal stability, while differential scanning calorimetry indicated an increase in the crystallinity of the films after the addition of PVDF-g-BT. Moreover, broadband dielectric spectroscopy revealed three dielectric processes, namely, glass–rubber relaxation (αa), relaxation associated with the polymer crystalline phase (αc), and slower relaxation in the nanocomposites resulting from the accumulation of charge on the interface between the PVDF-g-BT filler and the P(VDF-co-HFP) matrix. The dependence of the dielectric constant from the composition was analyzed, and we found that the highest permittivity enhancement was obtained by the highest concentration filler added to the largest concentration of P(VDF-co-HFP). Mechanical analysis revealed an improvement in Young’s modulus for all nanocomposites versus pristine P(VDF-co-HFP), confirming the uniformity of the distribution of the PVDF-g-BT nanocomposite with a strong interaction with the copolymer matrix, as also evidenced via scanning electron microscopy. The suggested system is promising for use in high-energy-density storage devices as supercapacitors.


Introduction
Dielectric nanocomposites have gained significant interest for their extensive applications in energy storage systems. These composites combine the high permittivity of perovskite oxides like BaTiO 3 with the desired characteristics of polymers, such as fracture toughness, flexibility, and ease of processing [1][2][3][4]. Specifically, ferroelectric fluorinated polymers have attracted considerable attention for high-tech applications. Examples include polyvinylidene fluoride (PVDF) and its copolymers, like poly(vinylidene fluorideco-trifluoroethylene) (P(VDF-co-TrFE)), poly(vinylidene fluoride-co-chlorotrifluoroethylene) (P(VDF-co-CTFE)), poly(vinylidene fluoride-co-hexafluoropropylene) (P(VDF-co-HFP)), and the terpolymer P(VDF-ter-TrFE-ter-CTFE). These polymers have drawn attention because, among all polymeric materials, they have the greatest dielectric constants (ε ∼10-12 at 1 kHz) [5][6][7][8][9][10][11], which are a result of their strong C-F dipole moment [12]. of BT were used for the preparation of our core-double-shell-structured samples. Such nanocomposites were produced through different loading (feed) of BT nanoparticles during production, namely 10% and 20% in weight; these fractions are used to label these nanocomposites in this work. However, the actual weight concentrations of BT nanoparticles grafted using PVDF chains were 16% and 38%, respectively, after the removal of ungrafted and physisorbed chains by washing them with acetone, as determined in a previous work by our team [40]. The HR MAS NMR 19 F spectra of the two PVDF-g-BT nanocomposites employed are reported in Figures S2 and S3 of the Supporting Information [35]. TEM pictures of the BT particles grafted with PVDF chains were reported in Ref. [35], giving evidence of the core-shell structure.

Film Preparation
The nanocomposite films were fabricated using the solution blending method with varying weight fractions of the components. The nomenclature employed in this study is presented in Table 1. For example, to prepare 10-10 nanocomposite films, 10 wt% (0.4 g) of the PVDF-g-BT nanocomposite, obtained from a 10% feed concentration, was dispersed in DMF (dimethylformamide) under ultrasonication for 4 h. Simultaneously, 90% by weight (3.6 g) of the P(VDF-co-HFP) copolymer was solubilized in DMF at room temperature for 4 h. Subsequently, the nanocomposite was slowly added to the copolymer solution while vigorously stirring the mixture at room temperature. The solution was prepared in an Erlenmeyer flask covered with a rubber septum. The resulting mixture was stirred for 24 h before being put into a Teflon mold and dried at 120 • C in an oven (Scheme 1). Table 1. Core-double-shell PVDF-g-BT x%/P(VDF-co-HFP) (X/Y) nanocomposite nomenclature. * x% is the weight (feed) ratio applied during the RAFT polymerization of VDF to produce PVDF-g-BT [35].

Materials
Commercially available P(VDF-co-HFP) (Kynar Flex-3120, 3 mol% of HFP) copolymer in pellets was purchased from Arkema, France. The 19 F NMR spectrum of this polymer is reported in Figure S1 of the Supporting Information. DMF solvent was provided by Fisher. The PVDF-g-BT nanocomposites used in the present study were synthesized via the reversible addition-fragmentation chain transfer (RAFT) polymerization of VDF in the presence of BT nanoparticles functionalized with a xanthate group (Scheme S1 in the Supporting Information). PVDF-g-BT nanocomposites with two different concentrations of BT were used for the preparation of our core-double-shell-structured samples. Such nanocomposites were produced through different loading (feed) of BT nanoparticles during production, namely 10% and 20% in weight; these fractions are used to label these nanocomposites in this work. However, the actual weight concentrations of BT nanoparticles grafted using PVDF chains were 16% and 38%, respectively, after the removal of ungrafted and physisorbed chains by washing them with acetone, as determined in a previous work by our team [40]. The HR MAS NMR 19 F spectra of the two PVDF-g-BT nanocomposites employed are reported in Figures S2 and S3 of the Supporting Information [35]. TEM pictures of the BT particles grafted with PVDF chains were reported in Ref. [35], giving evidence of the core-shell structure.

Film Preparation
The nanocomposite films were fabricated using the solution blending method with varying weight fractions of the components. The nomenclature employed in this study is presented in Table 1. For example, to prepare 10-10 nanocomposite films, 10 wt% (0.4 g) of the PVDF-g-BT nanocomposite, obtained from a 10% feed concentration, was dispersed in DMF (dimethylformamide) under ultrasonication for 4 h. Simultaneously, 90% by weight (3.6 g) of the P(VDF-co-HFP) copolymer was solubilized in DMF at room temperature for 4 h. Subsequently, the nanocomposite was slowly added to the copolymer solution while vigorously stirring the mixture at room temperature. The solution was prepared in an Erlenmeyer flask covered with a rubber septum. The resulting mixture was stirred for 24 h before being put into a Teflon mold and dried at 120 °C in an oven (Scheme 1). Scheme 1. Preparation method of the nanocomposite films, and picture of the produced films. Table 1. Core-double-shell PVDF-g-BT x%/P(VDF-co-HFP) (X/Y) nanocomposite nomenclature.
Sample Composition PVDF-g-BT x%/P(VDF-co-HFP) (X/Y) Code x-X x% = BT/VDF Feed Mass in PVDF-g-BT * X/Y Mass Ratio PVDF-g-BT x%/ P(VDF-co-HFP) Scheme 1. Preparation method of the nanocomposite films, and picture of the produced films.

. Fourier Transform Infrared (FT-IR) Spectroscopy
A Perkin Elmer Spectrum 1000 FT-IR spectrometer with a universal Attenuated Total Reflection (ATR) accessory was used to collect the infrared spectra. A total of 16 scans were averaged at a resolution of 2 cm −1 while scanning the samples in transmittance mode between 4000 and 400 cm −1 .

Thermogravimetric Analysis (TGA)
Thermogravimetric measurements were conducted using TA Instruments Q50 apparatus. The samples were subjected to heating under an air atmosphere, which consisted of a mixture of nitrogen and oxygen, at flow rates of 60 mL/min and 40 mL/min, respectively. The heating rate employed was 20 • C/min, starting from room temperature and reaching a maximum temperature of 700 • C.

Differential Scanning Calorimetry (DSC)
The films were subjected to the following heating and cooling cycles for DSC measurements using a Netzsch DSC 200 F3: initial heating from 20 • C to 200 • C at 20 • C/min; cooling from 200 to −70 • C at 20 • C/min; isothermal plateau at −70 • C for 10 min; second heating from −70 • C to 200 • C at 10 • C/min; and final cooling from 200 • C to 20 • C at 40 • C/min. The second heating/cooling cycles were used for the DSC thermograms that are provided in this study. Prior to analysis, the instrument's calibration with noble metals was verified using an indium sample (T m = 156.6 • C).
The area of the enthalpy peak and the point at the bottom of the valley with the greatest depth were used to calculate the melting temperatures and enthalpies (H m ), respectively. The crystallinity level of the polymers, χ, was determined using the formula below: where ∆H m is the melting enthalpy of the specimen under discussion, and ∆H c (104.7 J g −1 ) refers to the melting enthalpy of a 100% crystalline pure α-phase PVDF, which is taken to be identical to that of P(VDF-co-HFP) [5,7].

Scanning Electron Microscopy (SEM)
The microstructure of the PVDF-g-BT@P(VDF-co-HFP) nanocomposite films was examined using a scanning electron microscope (SEM) equipped with energy-dispersive mapping (EDS) capabilities (Zeiss HD15). To enhance conductivity, all samples were deposited onto a conductive adhesive carbon tape and subsequently coated with platinum. This preparation allowed for improved imaging and analysis of the samples.

Dielectric Measurements
A Novocontrol Technologies GmbH & Co. (Montabaur, Germany) Alpha Analyzer spectrometer outfitted with a Novocontrol Quatro nitrogen gas flow cryostat was used to perform broadband dielectric spectroscopy (BDS). The dielectric measurements were conducted on round film portions (such as the ones shown in Scheme 1), which were placed between two parallel electrodes in the active cell. The sample thickness was in the range of 0.16 to 0.26 mm. Measurements were performed under a sinusoidal applied voltage of 1.5 V. Collecting the dielectric response while increasing the frequency f of the imposed sinusoidal voltage in the range of~5·10 −2 Hz to 10 7 Hz, with logarithmic increments, at constant temperature, maintained for around 30 min, allowed us to obtain the isothermal spectra. The temperature range that was investigated was from −100 • C to 125 • C, with steps of 5 or 10 • C. Using an INSTRON 5566 tensile machine with a 2-kN load cell, the samples underwent uniaxial tensile testing at a crosshead speed of 2.5 mm/min. The stiffness of the films was determined by analyzing the stress-versus-strain curves within the linear region. Four rectangular specimens were used for each sample, with the film thickness ranging from 0.16 to 0.26 mm. Trendlines were employed to calculate the stiffness within a strain range of 0-2%.

Results and Discussion
Using FT-IR, SEM, TGA, and DSC, the produced PVDF-g-BaTiO 3 /P(VDF-co-HFP) films were thoroughly analyzed. The dielectric and mechanical properties were also investigated via BDS and tensile stress measurements, with a particular focus on the former ones, which are the most relevant for energy storage applications.

FT-IR Spectral Analysis
The FT-IR spectra of the purest P(VDF-co-HFP) copolymer and the nanocomposite films 10-10, 10-20, 20-10, and 20-20 are shown in Figure 1. The band seen in the spectra at about 1062 cm −1 relates to the CF 2 's symmetrical stretching mode. Additionally, the peaks found at 1170 cm −1 and 1400 cm −1 are the -C-F-groups inside the vinyl moiety's scissoring and bending vibrations, respectively [6,7,9,41]. The assignment of FTIR bands to PVDF crystalline phases is often a subject of debate. In this study, we followed reference [41], which identifies specific bands for the α-phase (763 cm −1 ) as well as the two electroactive phases, β (1275 cm −1 ) and γ (1234 cm −1 ). Notably, the band at 763 cm −1 is observed in the copolymer but disappears in all nanocomposites. The same applies to the bands at 795 cm −1 and 972 cm −1 , which are also characteristic of the crystalline α-phase. Both β-phase and γ-phase bands were already present in the copolymer (Figure 1b, bands "b" and "c") but with minimal intensity. In the nanocomposites, the increase in the γphase is more pronounced compared to that in the β-phase. However, due to the limited crystallinity of the nanocomposites, these bands remain weak. Also, the amorphous phase of P(VDF-co-HFP) is clear from the band at 876 cm −1 (band "g" in Figure 1). posed sinusoidal voltage in the range of ~5·10 −2 Hz to 10 7 Hz, with logarithmic increments, at constant temperature, maintained for around 30 min, allowed us to obtain the isothermal spectra. The temperature range that was investigated was from −100 °C to 125 °C, with steps of 5 or 10 °C.

Mechanical Analysis
Using an INSTRON 5566 tensile machine with a 2-kN load cell, the samples underwent uniaxial tensile testing at a crosshead speed of 2.5 mm/min. The stiffness of the films was determined by analyzing the stress-versus-strain curves within the linear region. Four rectangular specimens were used for each sample, with the film thickness ranging from 0.16 to 0.26 mm. Trendlines were employed to calculate the stiffness within a strain range of 0-2%.

Results and Discussion
Using FT-IR, SEM, TGA, and DSC, the produced PVDF-g-BaTiO3/P(VDF-co-HFP) films were thoroughly analyzed. The dielectric and mechanical properties were also investigated via BDS and tensile stress measurements, with a particular focus on the former ones, which are the most relevant for energy storage applications.

FT-IR Spectral Analysis
The FT-IR spectra of the purest P(VDF-co-HFP) copolymer and the nanocomposite films 10-10, 10-20, 20-10, and 20-20 are shown in Figure 1. The band seen in the spectra at about 1062 cm −1 relates to the CF2's symmetrical stretching mode. Additionally, the peaks found at 1170 cm −1 and 1400 cm −1 are the -C-F-groups inside the vinyl moiety's scissoring and bending vibrations, respectively [6,7,9,41]. The assignment of FTIR bands to PVDF crystalline phases is often a subject of debate. In this study, we followed reference [41], which identifies specific bands for the α-phase (763 cm −1 ) as well as the two electroactive phases, β (1275 cm −1 ) and γ (1234 cm −1 ). Notably, the band at 763 cm −1 is observed in the copolymer but disappears in all nanocomposites. The same applies to the bands at 795 cm −1 and 972 cm −1 , which are also characteristic of the crystalline α-phase. Both β-phase and γ-phase bands were already present in the copolymer (Figure 1b, bands "b" and "c") but with minimal intensity. In the nanocomposites, the increase in the γ-phase is more pronounced compared to that in the β-phase. However, due to the limited crystallinity of the nanocomposites, these bands remain weak. Also, the amorphous phase of P(VDF-co-HFP) is clear from the band at 876 cm −1 (band "g" in Figure 1). Upon the introduction of PVDF-g-BT nanocomposites into the P(VDF-co-HFP) matrix, the peaks associated with the α-phase structure (762, 794, and 972 cm −1 ) are eliminated. This shows that the inclusion of fillers in the P(VDF-co-HFP) matrix induces a transformation of the α-phase to another crystalline phase or to an amorphous state. These findings align with a study by Kumar et al. [42], where nanocomposite films were prepared using graphene oxide embedded in a P(VDF-co-HFP) matrix through a solution casting method. In their study, the FT-IR spectra also demonstrated the disappearance of characteristic peaks related to the α-phase structure in the nanocomposite. Moreover, the P(VDF-co-HFP) bands at 1400, 1069, and 876 cm −1 experienced slight shifts to 1402, 1071, and 875 cm −1 , respectively, in the composite samples. These shifts observed in PVDF-g-BT/P(VDF-co-HFP) composite materials indicate the occurrence of PVDF-P(VDF-co-HFP) polymer interactions within the composite system.

Scanning Electron Microscopy (SEM)
Energy-dispersive spectroscopy (EDS) mapping and scanning electron microscopy (SEM) techniques were used for investigating the film's morphology. The drying conditions used during preparation have a significant impact on the shape of the cast films [6,43]. In a work by Tian et al. [43], P(VDF-co-HFP) membrane material was created by dissolving P(VDF-co-HFP) pellets in a solvent under two different circumstances. The membranes that were dried under atmospheric conditions exhibited a porous structure, whereas those dried at 60 • C under vacuum showed a denser structure. In our investigation, the drying procedure was carried out under vacuum at 120 • C, producing dense sheets. The crosssection morphologies of the PVDF-g-BT/P(VDF-co-HFP) nanocomposite films and the pure P(VDF-co-HFP) copolymer are shown in Figure 2.
formation of the α-phase to another crystalline phase or to an amorphous state. These findings align with a study by Kumar et al. [42], where nanocomposite films were pre pared using graphene oxide embedded in a P(VDF-co-HFP) matrix through a solution casting method. In their study, the FT-IR spectra also demonstrated the disappearance o characteristic peaks related to the α-phase structure in the nanocomposite. Moreover, the P(VDF-co-HFP) bands at 1400, 1069, and 876 cm −1 experienced slight shifts to 1402, 1071 and 875 cm −1 , respectively, in the composite samples. These shifts observed in PVDF-g BT/P(VDF-co-HFP) composite materials indicate the occurrence of PVDF-P(VDF-co-HFP polymer interactions within the composite system.

Scanning Electron Microscopy (SEM)
Energy-dispersive spectroscopy (EDS) mapping and scanning electron microscopy (SEM) techniques were used for investigating the film's morphology. The drying conditions used during preparation have a significant impact on the shape of the cast films [6,43]. In a work by Tian et al. [43], P(VDF-co-HFP) membrane material was created by dissolving P(VDF-co-HFP) pellets in a solvent under two different circumstances. The membranes that were dried under atmospheric conditions exhibited a porous structure whereas those dried at 60 °C under vacuum showed a denser structure. In our investiga tion, the drying procedure was carried out under vacuum at 120 °C, producing dense sheets. The cross-section morphologies of the PVDF-g-BT/P(VDF-co-HFP) nanocomposite films and the pure P(VDF-co-HFP) copolymer are shown in Figure 2. There are many nanoparticle fillers in the polymer. The PVDF-g-BT nanocomposites are clearly well integrated into the P(VDF-co-HFP) copolymer in all samples. In addition no obvious aggregation, cavities, or flaws are seen in the films. These results suggest tha the grafted PVDF polymer shell significantly improves the adhesion between the BT na noparticles and the P(VDF-co-HFP) matrix, resulting in be er PVDF-g-BT nanoparticle dispersion. Qian et al. [32], Ma et al. [34], and Jiang et al. [37], who studied the morphology of fluorinated BT-g-styrene/P(VDF-TrFE-CTFE) nanocomposites, double-shell PVDF-g There are many nanoparticle fillers in the polymer. The PVDF-g-BT nanocomposites are clearly well integrated into the P(VDF-co-HFP) copolymer in all samples. In addition, no obvious aggregation, cavities, or flaws are seen in the films. These results suggest that the grafted PVDF polymer shell significantly improves the adhesion between the BT nanoparticles and the P(VDF-co-HFP) matrix, resulting in better PVDF-g-BT nanoparticle dispersion. Qian et al. [32], Ma et al. [34], and Jiang et al. [37], who studied the morphology of fluorinated BT-g-styrene/P(VDF-TrFE-CTFE) nanocomposites, double-shell PVDF-g-BT/PVDF composites, and P(VDF-co-HFP) composites with core-structured Fe 2 O 3 @BT nanofillers, provided the same findings.
Before SEM analysis, the PVDF-g-BT/P(VDF-co-HFP) core-double-shell nanocomposite films obtained after the casting were uniform, and no visible cracks were observed. One explanation could be that the electron-beam induces cracking in the composites' thin films due to the evaporation of remaining solvent, as observed by Schneider et al. [44]. In this study, an original suspension had a small amount of immiscible liquid added to it. This additional liquid keeps the particles' cohesiveness, changes the structure, speeds up drying, and reduces cracks. SEM was used to confirm the morphology before and after adding the immiscible liquid.
In addition, Yadavalli et al. [45] observed a curious cracking phenomenon in organicinorganic halide perovskite thin films during SEM. This team reported that this phenomenon can be seen in numerous published SEM micrographs. These films served as a demonstration of the mechanisms causing this e-beam-induced damage, obviating the need for thorough SEM characterization and comprehension. Tensile stresses accumulate because of the e-beam-induced fast volatilization of the organic species from the surface of these films in the SEM, and these stresses are controlled by the thin-film grain size to cause grain boundary cracking.
To look at the spatial distribution of the nanoparticle elements, the EDS method was used. The elemental mapping of the 10-10 film (PVDF-g-BT 10% @ P(VDF-co-HFP) (10/90)) is shown in Figure 3. According to the findings, the sample contains elements of Ba, Ti, C, O, and F, which are also present in core-double-shell nanocomposite films. Additionally, the existence of Ba, Ti, and O components suggests that BT-modified nanoparticles have dispersed.
This additional liquid keeps the particles' cohesiveness, changes the structure, speeds up drying, and reduces cracks. SEM was used to confirm the morphology before and after adding the immiscible liquid.
In addition, Yadavalli et al. [45] observed a curious cracking phenomenon in organicinorganic halide perovskite thin films during SEM. This team reported that this phenomenon can be seen in numerous published SEM micrographs. These films served as a demonstration of the mechanisms causing this e-beam-induced damage, obviating the need for thorough SEM characterization and comprehension. Tensile stresses accumulate because of the e-beam-induced fast volatilization of the organic species from the surface of these films in the SEM, and these stresses are controlled by the thin-film grain size to cause grain boundary cracking.
To look at the spatial distribution of the nanoparticle elements, the EDS method was used. The elemental mapping of the 10-10 film (PVDF-g-BT 10% @ P(VDF-co-HFP) (10/90)) is shown in Figure 3. According to the findings, the sample contains elements of Ba, Ti, C, O, and F, which are also present in core-double-shell nanocomposite films. Additionally, the existence of Ba, Ti, and O components suggests that BT-modified nanoparticles have dispersed.

Thermogravimetric Analysis (TGA)
The thermal stability of both the pristine P(VDF-co-HFP) copolymer and the nanocomposite films was investigated under ambient conditions using thermogravimetric analysis (TGA). The thermograms obtained are presented in Figure 4, while Table 2 provides a summary of the initial degradation temperature (Td) and weight losses at 650 °C.

Thermogravimetric Analysis (TGA)
The thermal stability of both the pristine P(VDF-co-HFP) copolymer and the nanocomposite films was investigated under ambient conditions using thermogravimetric analysis (TGA). The thermograms obtained are presented in Figure 4, while Table 2 provides a summary of the initial degradation temperature (T d ) and weight losses at 650 • C.     The pristine P(VDF-co-HFP) copolymer exhibited degradation starting from 446 • C, with a weight loss of 99.9% at 650 • C. Conversely, the PVDF-g-BT/P(VDF-co-HFP) films, prepared from 10-10, 10-20, 20-10, and 20-20 nanocomposites, demonstrated higher starting degradation temperatures, measuring 459 • C, 460 • C, 462 • C, and 464 • C, respectively, under similar conditions. In our case, with a moderate quantity of PVDF-g-BT-modified nanoparticles, the thermal stability of the double-shell nanocomposites was improved. Comparatively, an increase in T d from 13 to 18 • C was observed compared to the pristine P(VDF-co-HFP). This improvement in thermal stability can be attributed to the presence of well-dispersed PVDF-g-BT core nanocomposites in the P(VDF-co-HFP) matrix. Dispersed nanoparticles act as barriers, hindering the permeability of volatile degradation compounds, thus delaying the liberation of thermal degradation components compared with pristine polymers [46]. Furthermore, in nanocomposites with the same fraction of P(VDF-co-HFP) (e.g., 10-10 and 20-10 samples), an increase in decomposition temperature was observed for PVDF-g-BT fillers with a higher BaTiO 3 content (20%) compared to those with a lower fraction (10%). This increase can be attributed to improved dispersion and higher loading of the nanofiller. Previous studies conducted by our team [35] reported that the starting degradation temperatures of PVDF-g-BT 10% and PVDF-g-BT 20% were approximately 407 • C and 415 • C, respectively.

Differential Scanning Calorimetry (DSC)
DSC was employed to determine how the concentration of P(VDF-g-BT) nanocomposites affected the level of crystallinity and melting point of the nanocomposite films. The second heating and cooling DSC thermograms of our samples, obtained after a complete melting/recrystallization cycle, are presented in Figure 5a,b, respectively. The melting enthalpy (∆H m ), crystallization temperature (T c ), melting temperature (T m ), and crystallinity (χ) of each sample are summarized in Table 3. Thermograms are shifted vertically for clarity. Table 3. Melting enthalpy (ΔHm), crystallization temperature (Tc), melting temperature (Tm) and crystallinity (χ) of P(VDF-co-HFP) copolymer and its nanocomposite films.
Samples Thermograms are shifted vertically for clarity. Table 3. Melting enthalpy (∆H m ), crystallization temperature (T c ), melting temperature (T m ) and crystallinity (χ) of P(VDF-co-HFP) copolymer and its nanocomposite films.  Figure 5a shows a single endothermic melting peak for each film. As anticipated, the melting temperature (T m ) of the pristine P(VDF-co-HFP) copolymer is 167 • C, as the low HFP content of 3 mol% slightly reduces the melting temperature of the copolymer below the 170 • C observed for PVDF alone [47]. The melting enthalpy (∆H m ) increases from 19 J/g for the pristine P(VDF-co-HFP) copolymer to 22 J/g and 26 J/g when introducing 10 wt% and 20 wt% of P(VDF-g-BT) loaded with 20 wt% of BaTiO 3 , respectively.
Also, the crystallization temperature (T c ) of P(VDF-co-HFP) is 117 • C (Figure 5b), which rises to 125 • C for the 10-10 nanocomposite. Furthermore, the crystallinity of the films increases from 18% for the pristine copolymer to 25% after incorporating 20 wt% of PVDF-g-BT loaded with 20 wt% of BaTiO 3 . This suggests that the P(VDF-g-BT) core-shell nanofiller provides numerous heterogeneous nucleation sites within the P(VDF-co-HFP) matrix, thereby enhancing crystallinity. The increase in melting and crystallization temperatures upon introducing fillers into the P(VDF-co-HFP) matrix has been observed in previous studies [29,48,49]. For example, Zhang et al. [48] prepared core-shell-structured BaTiO 3 @polyalinine nanoparticles embedded in a P(VDF-co-HFP) matrix and noted an increase in T c for small filler amounts, along with a higher value of crystallinity. They explained this behavior as the interplay of two factors: (i) the introduction of nanofillers into the P(VDF-co-HFP) matrix provides additional nucleation sites, promoting crystallinity, and (ii) the fillers can impede the movement of P(VDF-co-HFP) chains, delaying the crystallization process and resulting in an observed increase in crystallization temperature.

Dielectric Properties
The molecular dynamics of both the P(VDF-co-HFP) copolymer matrix and its coredouble-shell nanocomposites were examined using broadband dielectric spectroscopy (BDS), which enables the characterization of their behavior with respect to temperature and frequency. This technique measures the electrical impedance and provides information about the dielectric permittivity, dielectric loss, and electrical modulus [50]. These parameters are crucial for assessing the enhancement of permittivity, conductivity, and other electrical characteristics relevant to the intended applications of the materials.
3.5.1. Dielectric Properties of P(VDF-co-HFP) Matrix Figure 6, which depicts changes in the dielectric permittivity (ε ) and dissipation factor (tan δ = ε /ε ) as a function of temperature and frequency, exemplifies the dielectric relaxation behavior of P(VDF-co-HFP).
Two distinct relaxation maxima are identified for P(VDF-co-HFP) that are close to and higher than the glass transition temperature (T g , around −36 • C as measured) (Figure 6a,b). Secondary relaxation, denoted as β relaxation, is the only one observable at low temperatures. Moreover, the dielectric relaxation arising from segmental motions within the amorphous phase is observed in the temperature range of −40 • C to around 0 • C and is referred to as primary or glass-rubber relaxation (α a ) [51]. This relaxation originates from the micro-Brownian motion of polymer chains, causing long-range dipole relaxation motions. When the temperature reaches the glass transition temperature, the dipoles become sufficiently mobile to realign themselves with the applied electric field, leading to an increase in permittivity. β and α a relaxation partially overlap, as reported in previous studies on PVDF [40]. Hence, the combined relaxations observable at lower temperatures are denoted here as β/α a .
These parameters are crucial for assessing the enhancement of permi ivity, conductivity, and other electrical characteristics relevant to the intended applications of the materials.
3.5.1. Dielectric Properties of P(VDF-co-HFP) Matrix Figure 6, which depicts changes in the dielectric permi ivity (ε′) and dissipation factor (tan δ = ε″/ε′) as a function of temperature and frequency, exemplifies the dielectric relaxation behavior of P(VDF-co-HFP). Two distinct relaxation maxima are identified for P(VDF-co-HFP) that are close to and higher than the glass transition temperature (Tg, around −36 °C as measured) (Figure 6a,b). Secondary relaxation, denoted as β relaxation, is the only one observable at low temperatures. Moreover, the dielectric relaxation arising from segmental motions within the amorphous phase is observed in the temperature range of −40 °C to around 0 °C and is referred to as primary or glass-rubber relaxation (αa) [51]. This relaxation originates from the micro-Brownian motion of polymer chains, causing long-range dipole relaxation motions. When the temperature reaches the glass transition temperature, the dipoles become sufficiently mobile to realign themselves with the applied electric field, leading to an increase in permi ivity. β and αa relaxation partially overlap, as reported in previous studies on PVDF [40]. Hence, the combined relaxations observable at lower temperatures are denoted here as β/αa.
The real part of the dielectric function, or permi ivity, ε′, has a point of inflection at the lowest frequencies, connected to an additional relaxation process, known as αc The real part of the dielectric function, or permittivity, ε , has a point of inflection at the lowest frequencies, connected to an additional relaxation process, known as α c relaxation. This process is related to relaxation occurring within the crystalline phase of P(VDF-co-HFP), and shows up at temperatures around 0 to 40 • C [51,52].
Furthermore, a distinct contribution from conductivity, σ dc , can be observed on the low-frequency side of the tan δ plot (Figure 6d). This conductivity is due to the drift of ionic impurities under the action of the applied electric field, leading to an upturn in the tan δ value at high temperatures, as visible, for instance, from the value of tan δ (10.5) at 0.1 Hz and 90 • C in Figure 6d. This conduction mechanism can give rise to a phenomenon of electrode polarization (EP) [53], as well as to interfacial polarization known as Maxwell-Wagner-Sillars (MWS) polarization, that occurs at nanoscale phase segregations such as the ones between amorphous and crystalline regions of the polymer [54], as well as between the polymer and nanoscale inclusions in nanocomposites. Conductivity effects are evidenced by a rapid increase in ε at low frequencies, as visible, for instance, in Figure 6c, where ε = 100 at 0.1 Hz and 90 • C. As mentioned above, this mechanism is associated with the presence of free charge carriers, typically ions, that can drift under the applied electric field toward the electrodes, and/or accumulate around the various interfaces present in the sample. Macroscopic polarization, which appears much larger than the polarization occurring in the bulk of the material, can result from such charge accumulation. We note that such a conductivity contribution can overlap with that of α c relaxation, making it difficult to discriminate the different relaxations. Figure 6a shows that in the glassy state, that is, below the glass transition temperature, the dipolar orientation is mostly impeded. The dielectric constant values of the P(VDF-co-HFP) copolymer at such temperatures are low, and rather independent of both temperature and frequency. In this case, high-frequency permittivity, ε ∞ , dominates the dielectric constant. For instance, ε ∞ ≈ 3.5 at 10 Hz and −100 • C, which is a common value for atomic and electronic polarization in polar materials. The dielectric permittivity of P(VDF-co-HFP) at temperatures higher than T g , that is, in the temperature range of −20 • C to 40 • C, increases with increasing temperature and decreasing frequency. At room temperature, it amounts to around 8 at 1 kHz (Figure 6a), which is consistent with the values known from the literature (ε ~10), depending on the HFP content in the P(VDF-co-HFP) copolymer [52,55,56]. This permittivity increase is attributed to the enhanced segmental motions of dipole units within P(VDF-co-HFP), associated with both α a and α c relaxation.
3.5.2. Dielectric Properties of Core-Double-Shell PVDF-g-BT/P(VDF-co-HFP) Nanocomposites The dielectric properties of the nanocomposites were also measured as a function of frequency and temperature, exhibiting similar trends across all samples. Figure 7 illustrates the variation in ε and tan δ of the 20-20 nanocomposite film with temperature at different frequencies (plots for samples 10-10, 10-20, and 20-10 can be found in Figure S4 of the Supporting Information). At low frequencies, the real part of permittivity (ε ) displays two inflection points associated with two relaxation processes: β/α a around the glass transition temperature (T g ) (−40 • C to 0 • C) and α c above T g (0 • C to 60 • C). These relaxations correspond to the dipole relaxation phenomena occurring in the amorphous and crystalline phases of the polymer matrix, respectively. Figure 7b presents the loss tangent (tan δ) as a function of temperature and frequency, with arrows indicating the evolution of these processes. The relaxation peak related to the β/α a process shifts to higher temperatures with increasing frequency. Notably, tan δ exhibits an inflection point corresponding to α c relaxation, which is partly masked by the conductivity contribution arising from electrode polarization and/or the interfacial Maxwell-Wagner-Sillars (MWS) relaxation occurring at high temperatures and lower frequencies. This "slow" process is responsible for the significantly higher values of ε observed at high temperatures and low frequencies (e.g., ε = 1000 at 0.1 Hz and 90 • C, Figure 7a). The dielectric permittivity increases with temperature for all samples. For instance, at 1 Hz and 60 • C, the dielectric permittivity increases to 20 for P(VDF-co-HFP) (Figure 6), while in the 20-20 nanocomposite, it reaches a value close to 40 (Figure 7a). This rapid increase in the dielectric constant with temperature is likely attributed to the enhanced interfacial polarization effect [54].    Figure S5 of the Supporting Information. The selected temperatures for these spectra are −20 °C, 40 °C, and 90 °C, which are near or above the glass transition temperature (Tg).  temperatures for these spectra are −20 • C, 40 • C, and 90 • C, which are near or above the glass transition temperature (T g ). Figure 7. Dielectric permi ivity, ε′ (a) and loss tangent, tan δ (b) of the 20-20 nanocomposite film versus temperature at different frequencies. The label αa indicates both β and αa relaxation (see text). Figure 8 presents selected representative isothermal spectra of ε′ and tan δ for the 20-20 nanocomposite film. The corresponding spectra for the 10-10, 10-20, and 20-10 nanocomposite films can be found in Figure S5 of the Supporting Information. The selected temperatures for these spectra are −20 °C, 40 °C, and 90 °C, which are near or above the glass transition temperature (Tg). Electrode and interfacial polarization resulting from conductivity dominate in the lower frequency range, leading to significantly high values of ε′. This phenomenon is particularly noticeable at higher temperatures, as observed in the 90 °C isotherm. Similar isotherms, however, exhibit a plateau accompanied by a decrease in ε′ at greater frequencies ( Figure 8a). Due to the polar segments' rotational motion within the material, which prevents them from following the applied electric field, the dielectric permi ivity decreases with increasing frequency.
At 90 °C, the plateau in ε′ is broad and centered around 200 kHz, which falls within the frequency range where conductivity effects are not prominent. In contrast, at lower temperatures, the same plateau shifts to lower frequencies. Additionally, a second step in ε′ with frequency appears, signifying the existence of αc relaxation. The findings reported for the reference polymer P(VDF-co-HFP) are consistent with this behavior. At −20 °C, Electrode and interfacial polarization resulting from conductivity dominate in the lower frequency range, leading to significantly high values of ε . This phenomenon is particularly noticeable at higher temperatures, as observed in the 90 • C isotherm. Similar isotherms, however, exhibit a plateau accompanied by a decrease in ε at greater frequencies ( Figure 8a). Due to the polar segments' rotational motion within the material, which prevents them from following the applied electric field, the dielectric permittivity decreases with increasing frequency.
At 90 • C, the plateau in ε is broad and centered around 200 kHz, which falls within the frequency range where conductivity effects are not prominent. In contrast, at lower temperatures, the same plateau shifts to lower frequencies. Additionally, a second step in ε with frequency appears, signifying the existence of α c relaxation. The findings reported for the reference polymer P(VDF-co-HFP) are consistent with this behavior. At −20 • C, large frequencies of β/α a relaxation are seen, but low frequencies of α c relaxation are observed within our spectral window. Figure 8b's representation of the plot of tan δ as a function of frequency might be used to gain a more thorough understanding, although with varied temperature dependencies, the contributions of β/α a and α c relaxation exhibit a trend towards lower frequencies as the temperature drops. As the glass transition temperature is approached, these relaxations are likely to combine. The significant increase in dielectric permittivity observed at 90 • C can be attributed to the presence of a slow process associated with enhanced interfacial polarization effects resulting from conduction. This effect primarily manifests in the low-frequency range, as interfacial polarization takes longer to develop compared to other types of processes. The existence of this slow process at low frequency and high temperature (T = 90 • C) is also shown in Figure 8b. The significant differential in electrical conductivity between the nanofillers and the copolymer matrix can be used to explain this phenomenon. It causes an increase in charge carriers at the interfaces where the different components of the nanocomposite film come together, which causes noticeable polarization and, as a result, a high dielectric constant [57]. Figure 9 compares the four core-double-shell-structured nanocomposites to the pristine P(VDF-co-HFP) copolymer, showing the frequency dependence of dielectric permittivity and dielectric loss tangent recorded at 20 • C. At this temperature, the value of the plateau in the frequency range of 10 2 to 10 5 Hz is due to the contribution of both the BT nanoparticle inclusion and the polymer (β and α a process), while the contribution of crystalline relaxation α c , as well as of the conductivity effects, starts at higher temperatures. We notice that the dielectric constant and the BT fraction do not show the same trend, because of the contribution of the amorphous polymer fraction, which should be determined quantitatively to rationalize the obtained results. However, a qualitative analysis can be conducted by assuming that the crystallinity of P(VDF-co-HFP) has similar values among the different core-double-shell nanocomposites, which should be acceptable as a first assumption, since it was observed via DSC that the crystallinity of fully melted and recrystallized nanocomposite samples ranged between 22% and 25%. In Table 4, the volume fractions φ in the core-double-shell nanocomposites of BT, of PVDF in PVDF-g-BT, and of P(VDF-co-HFP) are reported. Such fractions have been determined by solving the following linear equation system: where ρ BT = 6.02 g/cm 3 , and ρ P(VDF-co-HFP)~ρPVDF = 1.78 g/cm 3 .
nanoparticle inclusion and the polymer (β and αa process), while the contribution of crystalline relaxation αc, as well as of the conductivity effects, starts at higher temperatures. We notice that the dielectric constant and the BT fraction do not show the same trend, because of the contribution of the amorphous polymer fraction, which should be determined quantitatively to rationalize the obtained results. However, a qualitative analysis can be conducted by assuming that the crystallinity of P(VDF-co-HFP) has similar values among the different core-double-shell nanocomposites, which should be acceptable as a first assumption, since it was observed via DSC that the crystallinity of fully melted and recrystallized nanocomposite samples ranged between 22% and 25%. In Table 4, the volume fractions ϕ in the core-double-shell nanocomposites of BT, of PVDF in PVDF-g-BT, and of P(VDF-co-HFP) are reported. Such fractions have been determined by solving the following linear equation system: where BT = 6.02 g/cm 3 , and P(VDF-co-HFP) ~ PVDF = 1.78 g/cm 3 .  The highest value of dielectric constant is obtained for the 20-10 nanocomposite, which has an intermediate BT fraction (1.1 vol %, consistent with the BT fraction herein derived via TGA, that is, 1.3 vol %, within error; see Table 2), along with the highest fraction of P(VDF-co-HFP) (92.5 vol %). Since the amorphous fraction of P(VDF-co-HFP) is expected to be much higher than that of the PVDF fraction belonging to the PVDF-g-BT nanocomposite, which instead tends to be mostly crystalline [40], this result looks qualitatively consistent with expectations. For the same reasons, the lowest dielectric constant is obtained for the 10-10 sample, which has the smallest BT content (0.5 vol %), although the P(VDFco-HFP) fraction (91.0 vol %) is almost as high as for the 20-10 sample. Finally, for the other two samples (10-20 and 20-20), there is a balance between the effects of the two components (BT and amorphous polymer), which provides intermediate results for the dielectric constant. The rather high value of the dielectric constant of pristine P(VDF-co-HFP) compared to the core-double-shell nanocomposites could be ascribed to its smaller crystallinity value, due to the absence of crystallization sites provided by the addition of PVDF-g-BT to the nanocomposites. Similar non-monotonic trends of the dielectric constant with nanoparticle fractions have been observed in other studies [58][59][60]. For example, Khodaparast et al. [60] assembled nanocomposites by incorporating three different nanoparticles, namely, titania, silica, and alumina, into a PVDF matrix using a solution blending and casting method. These authors demonstrated that even when the fillers exhibited higher dielectric permittivity than PVDF alone, the resulting nanocomposites could exhibit a lower dielectric constant compared to the pure polymer. They highlighted the significance of the chemistry and interactions between the fillers and PVDF, which can play a more crucial role than the dielectric permittivity of the filler itself in determining the dielectric permittivity of the resulting nanocomposite.
Electrode and/or interface polarization can be the cause of the quick rise in permittivity at low frequencies. Other low-frequency relaxations may be completely eclipsed by this phenomenon (Figures 6-8) [61]. To address this issue, the concept of an electric modulus was introduced, which helps mitigate the influence of conductivity-induced polarization [61][62][63][64]. The definition of the electric modulus, abbreviated as M, is as follows: where the real and imaginary parts of the electric modulus, respectively, are denoted by M and M , and the real and imaginary parts of the dielectric function are denoted by ε and ε , respectively. To examine the bulk relaxation properties, the imaginary part of the electric modulus, sometimes referred to as Modulus or M , is frequently used in the form of loss curves [57]. This approach is favored because conductivity effects are observed as a dielectric peak rather than an increase at low frequencies. Figure 10 illustrates the variation in M with frequency at different temperatures (−20 • C, 40 • C, and 90 • C) near and above T g for both the P(VDF-co-HFP) matrix and the corresponding 20-10 nanocomposite. In Figure S6 of the Supporting Information, the dielectric loss modulus (M ) of P(VDF-co-HFP) and the nanocomposite films (10-10) is presented as a function of frequency in the temperature range of 0 • C to 100 • C.  A notable difference is observed between the P(VDF-co-HFP) copolymer and the nanocomposite films. In contrast to the copolymer, which shows two relaxation peaks, nanocomposite films show a third relaxation process, which is sometimes referred to as a "slow" process in line with earlier research [40,65]. The accumulation of free charges at the interfaces between the fillers and the polymer matrix [54,66,67] is what causes this "slow" process, which is frequently linked to interfacial polarization [65].
The M″-versus-frequency behavior of the 20-20 nanocomposite film at various temperatures is presented in Figure S7 of the Supporting Information. These findings are consistent with the matrix's and associated nanocomposites' dynamic molecular mobility, which was previously studied in terms of permi ivity and tan δ. Figure 11 provides a comparison of M″ between P(VDF-co-HFP) and its nanocompo- A notable difference is observed between the P(VDF-co-HFP) copolymer and the nanocomposite films. In contrast to the copolymer, which shows two relaxation peaks, nanocomposite films show a third relaxation process, which is sometimes referred to as a "slow" process in line with earlier research [40,65]. The accumulation of free charges at the interfaces between the fillers and the polymer matrix [54,66,67] is what causes this "slow" process, which is frequently linked to interfacial polarization [65].
The M -versus-frequency behavior of the 20-20 nanocomposite film at various temperatures is presented in Figure S7 of the Supporting Information. These findings are consistent with the matrix's and associated nanocomposites' dynamic molecular mobility, which was previously studied in terms of permittivity and tan δ. Figure 11 provides a comparison of M between P(VDF-co-HFP) and its nanocomposites at 40 • C and 90 • C, showcasing both the relaxation in the crystalline phase and the slow relaxation observed in the nanocomposites.
nanocomposite films show a third relaxation process, which is sometimes referred to as a "slow" process in line with earlier research [40,65]. The accumulation of free charges at the interfaces between the fillers and the polymer matrix [54,66,67] is what causes this "slow" process, which is frequently linked to interfacial polarization [65].
The M″-versus-frequency behavior of the 20-20 nanocomposite film at various temperatures is presented in Figure S7 of the Supporting Information. These findings are consistent with the matrix's and associated nanocomposites' dynamic molecular mobility, which was previously studied in terms of permi ivity and tan δ. Figure 11 provides a comparison of M″ between P(VDF-co-HFP) and its nanocomposites at 40 °C and 90 °C, showcasing both the relaxation in the crystalline phase and the slow relaxation observed in the nanocomposites. To perform a quantitative analysis of the dielectric relaxations, the Havriliak-Negami (HN) function is employed: To perform a quantitative analysis of the dielectric relaxations, the Havriliak-Negami (HN) function is employed: where the index k denotes the considered relaxation process (α c and α a ), f 0k is the related relaxation frequency, ∆ε k is the dielectric relaxation strength, defined as ∆ε = ε U − ε R (ε U and ε R are the low (unrelaxed)-and the high (relaxed)-frequency limits of the real part of dielectric permittivity with respect to the relaxation frequency range, in our case, ε R = ε ∞ ), and a k and b k are the parameters describing the distribution width and asymmetry of relaxations, respectively [52]. The conductivity contribution is also considered in this fitting function, where σ dc is the dc conductivity and n is a conductivity exponent to model different conduction mechanisms. All fittings were performed on the ε (dielectric loss) data to the imaginary part of Equation (4). The HN function is well suited to describing the dielectric relaxations of polymeric materials. The fitting procedure aims to determine appropriate values for the parameters f 0k , ∆ε k , a k , and b k by fitting the HN function to experimental data. The resulting values provide insights into the nature of the dielectric relaxation process and the underlying dynamics in the material being studied. Table 5 summarizes the fitting parameters for the α c relaxation process at T = 40 • C. This temperature was chosen because it is close enough to room temperature, and because an α c relaxation peak is exhibited at such temperature within the explored frequency window, in order to allow reasonable fitting for the estimation of the relaxation parameters. We notice that the α c relaxation-related increase in dielectric strength (∆ε) is higher in the nanocomposites than in the pristine P(VDF-co-HFP) matrix. The lowest increment of ∆ε is found for the 20-10 nanocomposite, that has the lowest PVDF crystalline fraction, derived as the sum of the grafted PVDF fraction (considered as 100% crystalline) and the crystalline fraction of P(VDF-co-HFP) (considered as around 25% crystallinity). This finding is consistent with the assumption that α c relaxation is related to the crystalline fraction. The width parameter a appears to be stable around 0.6, while the symmetry parameter b shows that the shape of relaxation in composites is more symmetric than in the pure polymer.
Concerning f 0 and ∆ε, the higher the crystalline volume fraction, the higher the dielectric relaxation strength associated with the α c process. Table 5. HN parameters for α c relaxation at T = 40 • C for pristine P(VDF-co-HFP) and its corresponding nanocomposite films.

Samples
Log f 0 ∆ε a b The temperature dependence of thermally activated processes is described by the Arrhenius relation: where f A is the Arrhenius frequency for infinite temperature, E a is the activation energy for the process, R is the gas constant, and T is the absolute temperature. This relationship is known to describe the relaxation frequencies of many dielectric processes. Figure 12 exhibits the logarithmic relaxation frequency (Log f 0 ) as a function of the inverse temperature (1000/T) for the α c and slow processes. This particular kind of Arrhenius plot is known as a relaxation plot. A relaxation process exhibiting a linear trend on a relaxation plot is referred to as having Arrhenius behavior. Both processes evidence an Arrhenius trend; therefore, fitting the data with the Arrhenius relationship of Equation (5) provides the Arrhenius frequency and the activation energy in each case. The results are summarized in Table 6. relaxation plot is referred to as having Arrhenius behavior. Both processes evidence an Arrhenius trend; therefore, fi ing the data with the Arrhenius relationship of Equation (5) provides the Arrhenius frequency and the activation energy in each case. The results are summarized in Table 6.  We can conclude that the P(VDF-co-HFP) copolymer exhibits slower relaxation, while nanocomposites with lower activation energies demonstrate faster relaxation processes, by comparing the activation energy values of the various samples.
We chose the data obtained at lower temperatures (−100 °C to −10 °C) and fi ed the dielectric loss data using Equation (4) for all samples in order to quantify β and αa relaxa-  We can conclude that the P(VDF-co-HFP) copolymer exhibits slower relaxation, while nanocomposites with lower activation energies demonstrate faster relaxation processes, by comparing the activation energy values of the various samples.
We chose the data obtained at lower temperatures (−100 • C to −10 • C) and fitted the dielectric loss data using Equation (4) for all samples in order to quantify β and α a relaxation. The outcomes are displayed in Figure 13. The obtained trend is approximately linear, denoting Arrhenius behavior, although with an inflection close to the expected glass transition temperature for all samples. We can conclude that the P(VDF-co-HFP) copolymer exhibits slower relaxation, while nanocomposites with lower activation energies demonstrate faster relaxation processes, by comparing the activation energy values of the various samples.
We chose the data obtained at lower temperatures (−100 °C to −10 °C) and fi ed the dielectric loss data using Equation (4) for all samples in order to quantify β and αa relaxation. The outcomes are displayed in Figure 13. The obtained trend is approximately linear, denoting Arrhenius behavior, although with an inflection close to the expected glass transition temperature for all samples. Figure 13. The logarithmic relaxation frequency variation as a function of inverse temperature for pristine P(VDF-co-HFP) copolymer and related nanocomposite films for the β process, with deviation from Arrhenius behavior due to the α a process (see text).
The Arrhenius parameters obtained by fitting the data from Figure 13 using Equation (5) are listed in Table 7. A lower Coefficient of Determination (COD) denotes higher deviation from the ideal Arrhenius behavior. The deviation occurs close to the expected glass transition temperature, likely caused by the influence of primary α a relaxation. This relaxation seems essentially unaffected by the fraction of nanocomposites. Table 7. Arrhenius parameters for β/α a relaxation, obtained via fitting using Equation (5). COD stands for Coefficient of Determination, which characterizes the quality of fitting.

Samples
Log f A E a (kJ mol −1 ) COD

Mechanical Properties
Through uniaxial tensile tests, the mechanical characteristics of the nanocomposite films were identified. The stress-versus-strain curves for both the pristine P(VDF-co-HFP) copolymer and the prepared nanocomposite films are depicted in Figure 14. The samples demonstrate ductile behavior, with the pristine copolymer exhibiting more pronounced ductility compared to the nanocomposite films, where the ductility is reduced.
During the testing procedure, four measurements were performed for each sample to accurately assess the key parameters defining the mechanical performance of the resulting films. The obtained values for Young's modulus and tensile strength have been compiled and are depicted in Figure 15.

Mechanical Properties
Through uniaxial tensile tests, the mechanical characteristics of the nanocomposite films were identified. The stress-versus-strain curves for both the pristine P(VDF-co-HFP) copolymer and the prepared nanocomposite films are depicted in Figure 14. The samples demonstrate ductile behavior, with the pristine copolymer exhibiting more pronounced ductility compared to the nanocomposite films, where the ductility is reduced. During the testing procedure, four measurements were performed for each sample to accurately assess the key parameters defining the mechanical performance of the resulting films. The obtained values for Young's modulus and tensile strength have been compiled and are depicted in Figure 15. The P(VDF-co-HFP) copolymer displayed a tensile strength of 22 MPa, accompanied by an elongation at break value of approximately 500%. In the case of the 10 wt% P(VDFg-BT) nanocomposite, incorporating 10 wt% of BaTiO3, the elongation at break decreased to approximately 100%, while the tensile strength increased to 24 MPa. As anticipated, a reduction in the elongation at break was observed in all the nanocomposite films compared to the pristine copolymer, in accordance with previous studies [68][69][70].
In terms of Young's modulus, the materials can be ranked in decreasing order as follows: 20-20 > 10-10 > 10-20 > 20-10 > P(VDF-co-HFP). The addition of P(VDF-g-BT) nanocomposites to the P(VDF-co-HFP) matrix resulted in increased stiffness of the resulting films. This result is consistent with the results obtained by Ponnamma et al. [71], who prepared P(VDF-co-HFP) nanocomposites with BaTiO3 and hexagonal boron nitride using a casting method. Their mechanical measurements demonstrated an enhancement in both Young's modulus and the tensile strength of the nanocomposites compared to the P(VDFco-HFP) copolymer, along with a decrease in elongation at break. A similar trend was observed by Tarhini et al. [72], who incorporated graphene nanoflakes into a P(VDF-co-HFP) matrix. They observed an improvement in Young's modulus for the nanocomposite films compared to the pristine copolymer film. This enhancement can be a ributed to the restriction of segmental movements of P(VDF-co-HFP) chains due to the introduction of nanoparticles, as well as the uniform distribution of nanofillers, which form strong bonds with the matrix.

Conclusions
A series of double-shell PVDF-g-BT/P(VDF-co-HFP) nanocomposite films were synthesized via a solution mixing method using a pristine P(VDF-co-HFP) copolymer and P(VDF-g-BT) nanofillers. The obtained films were fully characterized using techniques The P(VDF-co-HFP) copolymer displayed a tensile strength of 22 MPa, accompanied by an elongation at break value of approximately 500%. In the case of the 10 wt% P(VDF-g-BT) nanocomposite, incorporating 10 wt% of BaTiO 3 , the elongation at break decreased to approximately 100%, while the tensile strength increased to 24 MPa. As anticipated, a reduction in the elongation at break was observed in all the nanocomposite films compared to the pristine copolymer, in accordance with previous studies [68][69][70].
In terms of Young's modulus, the materials can be ranked in decreasing order as follows: 20-20 > 10-10 > 10-20 > 20-10 > P(VDF-co-HFP). The addition of P(VDF-g-BT) nanocomposites to the P(VDF-co-HFP) matrix resulted in increased stiffness of the resulting films. This result is consistent with the results obtained by Ponnamma et al. [71], who prepared P(VDF-co-HFP) nanocomposites with BaTiO 3 and hexagonal boron nitride using a casting method. Their mechanical measurements demonstrated an enhancement in both Young's modulus and the tensile strength of the nanocomposites compared to the P(VDFco-HFP) copolymer, along with a decrease in elongation at break. A similar trend was observed by Tarhini et al. [72], who incorporated graphene nanoflakes into a P(VDF-co-HFP) matrix. They observed an improvement in Young's modulus for the nanocomposite films compared to the pristine copolymer film. This enhancement can be attributed to the restriction of segmental movements of P(VDF-co-HFP) chains due to the introduction of nanoparticles, as well as the uniform distribution of nanofillers, which form strong bonds with the matrix.

Conclusions
A series of double-shell PVDF-g-BT/P(VDF-co-HFP) nanocomposite films were synthesized via a solution mixing method using a pristine P(VDF-co-HFP) copolymer and P(VDF-g-BT) nanofillers. The obtained films were fully characterized using techniques such as FT-IR, SEM, TGA, and DSC. These analyses confirmed the successful incorporation and effective dispersion of P(VDF-g-BT) nanocomposites within the P(VDF-co-HFP) matrix, leading to improved thermal properties of the system. Broadband dielectric spectroscopy (BDS) investigations unveiled the presence of three dipolar and interfacial relaxation phenomena, which contributed to the observed increase in permittivity. Notably, the highest increase in the dielectric constant was achieved when utilizing the highest BT content in the filler combined with the lowest filler fraction. Mechanical analysis exhibited an improvement in Young's modulus and the tensile strength of the nanocomposite films versus the pristine copolymer. Consequently, the films exhibited increased stiffness, further validating the beneficial impact of the PVDF-g-BT nanocomposites on the mechanical properties of the system.
The concentration of core-shell structures is proportional to the fraction of BT, while the volume of the PVDF shell is proportional to the fraction of PVDF. The trends of the various physical quantities investigated could be compared to either one or the other. For instance: (i) The starting degradation temperature determined via TGA increases with the concentration of core-shell structures. (ii) Within an acceptable error, Young's modulus is higher for all nanocomposites than for the polymer alone and rises with the PVDF shell volume. (iii) Tensile strength cannot be assigned a trend, because the error bars are comparable to the dispersion of data. (iv) The elongation at break decreases with the core-shell structure concentrations.
Thanks to such interesting properties, these nanocomposite films have promising applications. One of the most suitable options, specifically, is to utilize them as electrolytes and binders for lithium-ion batteries [39,73].